ELEMENTS; The whole Fifteen BOOKS WITH ARCHIMEDES Theorems Of the Sphere and Cylinder, investi- By ISAAC BARROW, D. D. Late Master To which is added in this Edition, EUCLIDE'S DATA And a Brief TREATISE of Καθαρμοί ψυχῆς λογικῆς εἰσὶν οἱ μαθηματικαὶ όλιςήμαι. LONDON: Printed and Sold by W. Redmayne To the READER. F you are defirous, Courteous Reader, to know what according to the nature of the Work. Ibave endeavor'd to attain two ends chiefly; the first, to be very perspicuous, and at the same time so very brief, that. the Book may not swell to such a Bulk, as may be troublesome to carry about one, in which I think I have Succeeded, unless in my absence the Printer's care should frustrate my Design. Some of a brighter Genius, and endued with greater Skill, may have demonstrated most of these Propofitions with more nicety, but perhaps none with more fuccinctness than I have; especially fince I alter'd nothing in the number and order of the Author's Propofitions; nor presum'd either to take the liberty of rejecting, as less necessary, any of them, or of reducing Some of the easier fort into the rank of Axioms, as several bave done, and among others, that most expert Geometrician A. Tacquetus C. (whom I the more willingly name, because I think it is but civil to acknowledge that I have imitated him in some Points) after whose most acurate Edition I had no Thoughts of attempting any thing of this nature, till I confider'd that this most learned Man thought fit to publish only eight of Euclide's Books, which he took the pains to explain and embellish, having in a manner rejected and undervalued the other seven, as less appertaining to the Elements of Geometry. But my Province was origi a 2 : originally quite different, not that of writing the Elements of Geometry, after what method soever I pleas'd, but of demonstrating, in as few words as possible I cou'd, the whole Works of Euclide. As to four of the Books, viz. the seventh, eighth, ninth, and tenth, altho they don't so nearly appertain to the Elements of plain and solid Geometry, as the fix precedent and the tawo subsequent, yet none of the more skilful Geometricians can be so ignorant as not to know that they are very useful for Geometrical matters, not only by rea Son of the mighty near affinity that is between Arithmetick and Geometry, but also for the knowledge of both measurable and unmeasurable Magnitudes, so exseeding necessary for the Doctrine of both plain and folid figures. Now the noble Contemplation of the five Regular Bodies that is contain'd in the three last Books, cannot without great Injustice be pretermitted, fince that for the fake thereof our sοιχειωτής, being a Philosopher of the Platonic Sect, is said to have compos'd this universal System of Elements; as Proclus lib. 2. witnesseth in these Words, Ὅθεν δὴ κὶ τῆς συμπάσης σοιχειώσεως τέλο προιςἤσαλο τὴν τε καλυμθύων πλατωνικῶν χημάτων σύςασιν. Besides, I easily perswaded my self to think, that it would not be unacceptable to any Lover of these Sciences to bave in bis Poffeffion the whole Euclidean Work, as it is commonly cited and celebrated by all Men. Wherefore I resolu'd to omit no Book or Propofition of those that are found in P. Herigonius's Edition, whose Steps I was oblig'd closely to follow, by reason I took a Resolution to make use of most of the Schemes of the said Book, very well foreseeing that time would not allow me to form new 1 new ones, the sometimes I chose rather to do it. For the same reason I was willing to use for the most part Euclide's own Demonstrations, having only express'd them in a more succinct Form, unless perhaps in the second, thirteenth, and very few in the seventh, eighth, and ninth Book, in which it seem'd not worth my while to deviate in any particular from bim. Therefore I am not without good hopes that as to this part I bave in some measure satisfied both my own Intentions, and the Defire of the Studious. As for some certain Problems and Theorems that are added in the Scholions (or short Expositions) either appertaining (by reason of their frequent Use) to the nature of these Elements, or conducing to the ready Demonstration of those things that follow, or which do intimate the reasons of fome principal Rules of practical Geometry, reducing them to their original Fountains, these I say, will not, I bope, make the Book swell to a Size beyond the defigr'd Proportion. The other Butt, which I levell'd at, is to content the Desires of those who are delighted more with symbolical than verbal Demonstrations. In which kind, whereas most among us are accustom'd to the Symbols of Gulielmus Oughtredus, I therefore thought beft to make use, for the most part, of his.. None hitherto (as I know of) has attempted to interpret and publish Euclide after this manner, except P. Herigonius; whose Method (the indeed most excellent in many things, and very well accommodated for the particular purpose of that most ingenious Man) yet seems in my Opinion to labour under a double Defect. First, in regard that, altho of two or more Propositions, produ ced 1 |